Question

How do you factor given that `f(-14)=0` and `f(x)=x^3+11x^2-150x-1512`?

Answer 1

`(x-12)(x+9)(x+14)`

Explanation:

When you know that `f(c)=0` for some value `c`, you knot that `f(x)` has a factor `(x-c)`. So, you can divide `x^3+11x^2-150x-1512` by `(x+14)` via long division, or whatever method you prefer, and obtain

`(x^3+11x^2-150x-1512)/(x+14) = x^2-3 x-108`

which means

`(x^2-3 x-108)(x+14) = x^3+11x^2-150x-1512`

Again, `x^2-3 x-108` has solutions `12` and `-9`, which means that it can be written as `(x-12)(x+9)`

Thus, we can write

`x^3+11x^2-150x-1512 = (x^2-3 x-108)(x+14) = (x-12)(x+9)(x+14)`